Author Archives: Fabrice Baudoin

In the next few lectures, we will show that the diffusion semigroups theory we developed may actually be extended without difficulties to a manifold setting. As a motivation, we start with a very simple example. We first study the heat … Continue reading →

In the previous lectures, we have seen that if L is an essentially self-adjoint diffusion operator with respect to a measure, then by using the spectral theorem one can define a self-adjoint strongly continuous contraction semigroup on L2 with generator … Continue reading →

In this Lecture, we use the local regularity theory of subelliptic operators, to prove the existence of heat kernels. Proposition: Let be a locally  subelliptic diffusion operator with smooth coefficients that is essentially self-adjoint with respect to a measure . Denote … Continue reading →

Exercise: Show that if is the Laplace operator on , then for ,   Exercise: Let be an essentially self-adjoint diffusion operator on . Show that if the constant function and if , then   Exercise: Let be an essentially self-adjoint diffusion operator … Continue reading →

For geometric purposes, it is often very useful to use the language of vector fields to study diffusion operators. Let be a non-empty open set. A smooth vector field   on is a smooth map We will often regard a … Continue reading →

In this lecture, we show that the diffusion semigroup that was constructed in the previous lectures appears as the solution of a parabolic Cauchy problem. Under an ellipticity and completeness assumption, it is moreover the unique square integrable solution. Proposition: Let be an … Continue reading →